Natural superoscillations in monochromatic waves in D dimensions

For monochromatic waves satisfying the Helmholtz equation with wavenumber k(0), superoscillations correspond to local wavenumbers (magnitude of phase gradient) greater than k(0). Large values of local wavenumber are associated with phase singularities. For isotropic random waves (superpositions of many nonevanescent waves) in D dimensions, we show that the probability that a point in the field is superoscillatory increases from 0.293 to 0.394 as D increases from 1 to infinity. The peculiar case D = 1 is examined in detail.